{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/globally-convergent-newton-methods-for-ill","title":"Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses","arxiv_id":"1907.01771","date":"2019-07-03","proceeding":"NeurIPS 2019 12","authors":["Ulysse Marteau-Ferey","Francis Bach","Alessandro Rudi"],"abstract":"In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr{\\\"o}m projection techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(ndf $\\lambda$), a memory complexity of order O(df 2 $\\lambda$) and no dependence on the condition number, generalizing the results known for least-squares regression. Here n is the number of observations and df $\\lambda$ is the associated degrees of freedom. In particular, this is the first large-scale algorithm to solve logistic and softmax regressions in the non-parametric setting with large condition numbers and theoretical guarantees.","url_abs":"https://arxiv.org/abs/1907.01771v2","url_pdf":"https://arxiv.org/pdf/1907.01771v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"globally-convergent-newton-methods-for-ill","repo_url":"https://github.com/umarteau/Newton-Method-for-GSC-losses-","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}},{"paper_slug":"globally-convergent-newton-methods-for-ill","repo_url":"https://github.com/EigenPro/EigenPro","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[{"task_slug":"generalization-bounds","task_name":"Generalization Bounds"},{"task_slug":"regression-1","task_name":"regression"}],"methods":[{"method_slug":"logistic-regression","method_name":"Logistic Regression"},{"method_slug":"softmax","method_name":"Softmax"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1907.01771","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}